Adhesion of elastic spheres

Bradley (1932) showed that if two rigid spheres of radii R-1 and R-2 are placed in contact, they will adhere with a force 2 pi R Delta gamma, where R is the equivalent radius R1R2/(R-1 + R-2) and Delta gamma is the surface energy or 'work of adhesion' (equal to gamma(1) + gamma(2) - gamma(12)). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy -pi a(2) Delta gamma) how the Hertz equations for the contact of elastic spheres are modified by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)pi R Delta gamma, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley's answer. The discrepancy was explained by Tabor (1977), who identified a parameter mu = R(1/3)Delta gamma(2/3)/E*(2/3)epsilon governing the transition from the Bradley pull-off force 2 pi R Delta gamma to the JKR value (3/2)pi R Delta gamma. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard-Jones law of force between surfaces with the elastic equations for a halfspace), and confirmed that Tabor's parameter does indeed govern the transition. The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load-approach curves become S-shaped for values of mu greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of mu of 3 or more, but for low values of mu the simple Bradley equation better describes the behaviour under negative loads.